ht2
DESCRIPTION
Heat TransferTRANSCRIPT
ConvectionFree or natural convection (induced by buoyancy forces)
forced convection (driven externally)
May occur with phase change (boiling, condensation)
Convection
Heat transfer rate q = h( Ts-T )W
h=heat transfer coefficient (W /m2K)
(h is not a property. It depends on geometry ,nature of flow, thermodynamics properties etc.)
• Nusselt No. Nu = hx / k = (convection heat transfer strength)/ (conduction heat transfer strength)
• Prandtl No. Pr = / = (momentum diffusivity)/ (thermal diffusivity)
• Reynolds No. Re = U x / = (inertia force)/(viscous force)
Viscous force provides the dampening effect for disturbances in the fluid. If dampening is strong enough laminar flow
Otherwise, instability turbulent flow critical Reynolds number
Laminar Turbulent
Forced convection: Non-dimensional groupings
•Fluid particle adjacent to the solid surface is at rest.
•These particles act to retard the motion of adjoining layers.
FORCED CONVECTION:
External flow (Over flat plate)An internal flow is surrounded by solid boundaries that can restrict the development of its boundary layer, for example, a pipe flow. An external flow, on the other hand, are flows over bodies immersed in an unbounded fluid so that the flow boundary layer can grow freely in one direction. Examples include the flows over airfoils, ship hulls, turbine blades, etc.
laminar turbulenttransition
Dye streak
U U U
U
PAGE 4
• Boundary layer growth: x• Initial growth is fast• Growth rate d/dx 1/x, decreasing downstream.
• Wall shear stress: w 1/x• As the boundary layer grows, the wall shear stress decreases as the velocity gradient at the wall becomes less steep.
Laminar Boundary Layer Development
Boundary layer equations (laminar flow)
T
U
WT
T
U
x
y
• Equations for 2D, laminar, steady boundary layer flow
y
T
yy
Tv
x
Tu
y
u
ydx
dUU
y
uv
x
uu
y
v
x
u
:energyofonConservati
:momentum-xofonConservati
0:massofonConservati
• Note: For a flat plate, 0hence,constantis dx
dUU
Exact solutions: Blasius
31
21
31
21
PrRe664.0numberNusseltAverage
PrRe332.0numberNusseltLocal
ReRe
328.11t,coefficiendragAverage
,Re
Re
664.0tcoefficienfrictionSkin
Re
5
x knesslayer thicBoundary
0
0
221
L
xx
L
L
L
fD
y
wx
x
wf
x
uN
Nu
LUdxC
LC
y
uxU
UC
Heat transfer coefficient
• Local heat transfer coefficient:
x
k
x
kNuh xxx
31
21
PrRe332.0
• Average heat transfer coefficient:
L
k
L
kuNh L
31
21
PrRe664.0
• Film temperature, Tfilm
For heated or cooled surfaces, the thermophysical properties within the boundary layer should be selected based on the average temperature of the wall and the free stream;
TTT wfilm 21
Internal Flow Convection-constant surface temperature case
Another commonly encountered internal convection condition is when the surface temperature of the pipe is a constant. The temperature distribution in this case is drastically different from that of a constant heat flux case. Consider the following pipe flow configuration:
Tm,i Tm,o
Constant Ts
Tm Tm+dTm
qs=hA(Ts-Tm)
p
p
s
p s
Energy change mC [( ) ]
mC
Energy in hA(T )
Energy change energy in
mC hA(T )
m m m
m
m
m m
T dT T
dT
T
dT T
dx
Temperature distribution
p s
s m
m
mC hA(T ),
Note: q hA(T ) is valid locally only, since T is not a constant
, where A Pdx, and P is the perimeter of the pipe(T )
Integrate from the inlet to a diatance x downstr
m m
m
m
s P
dT T
T
dT hA
T mC
,
,
( )
0 0m
( )m
0
eam:
(T )
ln(T ) | , where L is the total pipe length
and h is the averaged convection coefficient of the pipe between 0 & x.
1, or
m
m i
m
m i
T x x xm
Ts P P
T xs T
P
x
dT hP Pdx hdx
T mC mC
PhT x
mC
h hdx hdxx
0
xhx
Temperature distribution
,
( )exp( ), for constant surface temperaturem s
m i s P
T x T Phx
T T mC
T x( )
x
Tm(x)
Constant surface temperatureTs
The difference between the averaged fluid temperature and the surface temperature decreases exponentially further downstream along the pipe.
Log-Mean Temperature Difference
,
,
, , , ,
For the entire pipe:
( )exp( ) or
ln( )
( ) (( ) ( ))
( )ln( )
where is calln( )
m o s o sP
om i s i P
i
P m o m i P s m i s m o
o iP i o s s lm
o
i
o ilm
o
i
T T T hAh PLmC
TT T T mCT
q mC T T mC T T T T
T TmC T T hA hA T
T
T
T TT
T
T
led the log mean temperature difference.
This relation is valid for the entire pipe.
Free Convection
A free convection flow field is a self-sustained flow driven by the presence of a temperature gradient. (As opposed to a forced convection flow where external means are used to provide the flow.) As a result of the temperature difference, the density field is not uniform also. Buoyancy will induce a flow current due to the gravitational field and the variation in the density field. In general, a free convection heat transfer is usually much smaller compared to a forced convection heat transfer. It is therefore important only when there is no external flow exists.
hot
cold
T
Flow is unstable and a circulatorypattern will be induced.
Basic Definitions
Buoyancy effect:
Warm,
Surrounding fluid, cold,
Hot plateNet force=(- gV
The density difference is due to the temperature difference and it can be characterized by ther volumetric thermal expansion coefficient, :
1 1 1( )PT T T T
T
Grashof Number and Rayleigh Number
Define Grashof number, Gr, as the ratio between the buoyancy force and the viscous force: 33
2 2
( )Sg T T Lg TLGr
• Grashof number replaces the Reynolds number in the convection correlation equation. In free convection, buoyancy driven flow sometimes dominates the flow inertia, therefore, the Nusselt number is a function of the Grashof number and the Prandtle number alone. Nu=f(Gr, Pr). Reynolds number will be important if there is an external flow. (combined forced and free convection.
• In many instances, it is better to combine the Grashof number and the Prandtle number to define a new parameter, the Rayleigh number, Ra=GrPr. The most important use of the Rayleigh number is to characterize the laminar to turbulence transition of a free convection boundary layer flow. For example, when Ra>109, the vertical free convection boundary layer flow over a flat plate becomes turbulent.